Discrete Search Space Research Articles

BMNABC: Binary Multi-Neighborhood Artificial Bee Colony for High-Dimensional Discrete Optimization Problems

Many meta-heuristic algorithms have been proposed to solve continuous optimization problems. Hence, researchers have applied various techniques to change these algorithms for discrete search spaces. Artificial bee colony (ABC) algorithm is one of the well-known algorithms for real search spaces. ABC has a good ability in exploration but it is weak in exploitation. Several binary versions of ABC have been proposed so far. Since the methods are based on the standard ABC, they have the disadvantage of ABC. In this article, a new binary ABC called binary multi-neighborhood ABC (BMNABC) has been introduced to enhance the exploration and exploitation abilities in the phases of ABC. BMNABC applies the near and far neighborhood information with a new probability function in the first and second phases. A more conscious search than the standard ABC is done in the third phase for those solutions which have been not improved in the previous phases. The performance of algorithm has been evaluated by low- and high-dimensional functions and the 0-1 multidimensional knapsack problems. The proposed method has been compared with state-of-the-art algorithms. The results showed that BMNABC had a better performance in terms of solution accuracy and convergence speed.

The ( $$1+\lambda $$ 1 + λ ) Evolutionary Algorithm with Self-Adjusting Mutation Rate

We propose a new way to self-adjust the mutation rate in population-based evolutionary algorithms in discrete search spaces. Roughly speaking, it consists of creating half the offspring with a mutation rate that is twice the current mutation rate and the other half with half the current rate. The mutation rate is then updated to the rate used in that subpopulation which contains the best offspring. We analyze how the $$(1+\lambda )$$ evolutionary algorithm with this self-adjusting mutation rate optimizes the OneMax test function. We prove that this dynamic version of the $$(1+\lambda )$$ EA finds the optimum in an expected optimization time (number of fitness evaluations) of $$O(n\lambda /\log \lambda +n\log n)$$ . This time is asymptotically smaller than the optimization time of the classic $$(1+\lambda )$$ EA. Previous work shows that this performance is best-possible among all $$\lambda $$ -parallel mutation-based unbiased black-box algorithms. This result shows that the new way of adjusting the mutation rate can find optimal dynamic parameter values on the fly. Since our adjustment mechanism is simpler than the ones previously used for adjusting the mutation rate and does not have parameters itself, we are optimistic that it will find other applications.

Binary dragonfly optimization for feature selection using time-varying transfer functions

The Dragonfly Algorithm (DA) is a recently proposed heuristic search algorithm that was shown to have excellent performance for numerous optimization problems. In this paper, a wrapper-feature selection algorithm is proposed based on the Binary Dragonfly Algorithm (BDA). The key component of the BDA is the transfer function that maps a continuous search space to a discrete search space. In this study, eight transfer functions, categorized into two families (S-shaped and V-shaped functions) are integrated into the BDA and evaluated using eighteen benchmark datasets obtained from the UCI data repository. The main contribution of this paper is the proposal of time-varying S-shaped and V-shaped transfer functions to leverage the impact of the step vector on balancing exploration and exploitation. During the early stages of the optimization process, the probability of changing the position of an element is high, which facilitates the exploration of new solutions starting from the initial population. On the other hand, the probability of changing the position of an element becomes lower towards the end of the optimization process. This behavior is obtained by considering the current iteration number as a parameter of transfer functions. The performance of the proposed approaches is compared with that of other state-of-art approaches including the DA, binary grey wolf optimizer (bGWO), binary gravitational search algorithm (BGSA), binary bat algorithm (BBA), particle swarm optimization (PSO), and genetic algorithm in terms of classification accuracy, sensitivity, specificity, area under the curve, and number of selected attributes. Results show that the time-varying S-shaped BDA approach outperforms compared approaches.

A splitting algorithm for simulation-based optimization problems with categorical variables

ABSTRACTIn the design of complex products, some product components can only be chosen from a finite set of options. Each option then corresponds to a multidimensional point representing the specifications of the chosen components. A splitting algorithm that explores the resulting discrete search space and is suitable for optimization problems with simulation-based objective functions is presented. The splitting rule is based on the representation of a convex relaxation of the search space in terms of a minimum spanning tree and adopts ideas from multilevel coordinate search. The objective function is underestimated on its domain by a convex quadratic function. The main motivation is the aim to find—for a vehicle and environment specification—a configuration of the tyres such that the energy losses caused by them are minimized. Numerical tests on a set of optimization problems are presented to compare the performance of the algorithm developed with that of other existing algorithms.

Discrete Multi-Valued Search Space Algorithm Based on Cockroach Swarms

Discrete Multi-Valued Search Space Algorithm Based on Cockroach Swarms

RETRACTED: A hybrid whale optimization algorithm based on local search strategy for the permutation flow shop scheduling problem

Abstract The flow shop scheduling problem is one of the most important types of scheduling with a large number of real-world applications. In this paper, we propose a new algorithm that integrates the Whale Optimization Algorithm (WOA) with a local search strategy for tackling the permutation flow shop scheduling problem. The Largest Rank Value (LRV) requires the algorithm to deal with the discrete search space of the problem. The diversity of candidate schedules is improved using a swap mutation operation as well. In addition to the insert-reversed block operation is adopted to escape from the local optima. The proposed hybrid whale algorithm (HWA) is incorporated with Nawaz–Enscore–Ham (NEH) to improve the performance of the algorithm. It is observed that HWA gives competitive results compared to the existing algorithms.

Convertible furniture design

Convertible furniture is used widely because of its multi-purpose and space-saving properties, but substantial professional skills are required to design practical convertible furniture. In this study, we introduce a system that allows unskilled users to design convertible furniture objects. By using a set of desired furniture units as inputs, our system generates a group of suggested convertible furniture objects, which can collapse into a compact form. The geometric problem involves connecting the furniture units via junctions so they occupy the smallest space in their compact form while still fulfilling the desired function in the expanded form. This optimization problem is non-trivial because of the very large discrete search space of the connected structures. Furthermore, there is a huge continuous space of geometric parameters where the connected units make contact due to the complex dependencies of their movements. We propose an efficient algorithm based on a tree selection approach and pre-computed junction-dependent configuration spaces to accelerate the optimization process in a high-dimensional, mixed discrete–continuous search space. We evaluated our system based on a wide range of inputs and a series of convertible furniture objects were generated. The experimental results demonstrate that our system can provide design suggestions as well as inspiring creativity among users.

A modified flower pollination algorithm for the multidimensional knapsack problem: human-centric decision making

In this paper, a new modified version of the flower pollination algorithm based on the crossover for solving the multidimensional knapsack problems called (MFPA) is proposed. MFPA uses the sigmoid function as a discretization method to deal with the discrete search space. The penalty function is added to the evaluation function to recognize the infeasible solutions and assess them. A two-stage procedure is called FRIO is used to treat the infeasible solutions. MFPA uses an elimination procedure to decrease any duplication in the population in order to increase the diversity. The proposed algorithm is verified on a set of benchmark instances, and a comparison with other algorithms available in literature is shown. Several statistical and descriptive analysis was done such as recoding the results of the best, mean, worst, standard deviation, success rate, and time to prove the effectiveness and robustness of MFPA. The empirical results show that the proposed algorithm can be an effective algorithm as human-centric decision-making model for solving the multidimensional knapsack problems.

Evolutionary computation, Winter 2016, vol. 24

In combinatorial optimization, the goal is to find an optimal solution, according to some objective function, from a discrete search space. These problems arise widely in industry and academia and, unfortunately, many of them are NP-hard and no polynomial time algorithm can guarantee their solution to a certified optimality unless P=NP. Therefore, in the last decades researchers have investigated the use of stochastic search algorithms to find near optimal solutions to these problems. In particular, great research efforts have been devoted to the development and application of metaheuristic algorithms to solve combinatorial optimization problems This special issue contains six high-quality articles addressing practical applications and theoretical developments of metaheuristic algorithms in the context of combinatorial optimization problems. The articles in this issue have been selected from among 23 submissions after a thorough peer review process. Their contents, outlined in the next paragraphs, reflect the diversity of the application domains and the methods applied to solve the problems.

Mixed Approaches to Handle Limitations and Execute Mutation in the Genetic Algorithm for Truss Size, Shape and Topology Optimization

A high-performance genetic algorithm for the optimal synthesis of trusses in discrete search spaces is developed. The main feature of the proposed computational procedure is the possibility of obtaining effective solutions without the violation of any constraint. In general, a varying of cross-sectional areas of bars, coordinates of nodes and topology system is provided. A group of individuals in the population can be accepted for further consideration only if all specified limitations have been fulfilled. Penalties that significantly change an objective function are introduced for other individuals. This mechanism of handling limitations provides for correction of inaccuracies that can introduce penalty functions for satisfying the problem conditions. Both a random change to the entire set of admissible values and a random choice of values among adjacent elements in this set can be performed during the mutation stage. Standard test examples for benchmark mathematical functions and trusses show high efficiency of the considered iterative procedure in terms of solution accuracy.

A comparison of different transfer functions for binary version of grey wolf optimiser

Grey Wolf Optimiser (GWO) is the recently proposed meta-heuristic algorithm inspired by grey wolves. The original version of GWO has been proposed to solve problems in continuous search spaces. However, there are many optimisation problems in discrete binary search spaces such as feature selection and dimensionality reduction. We applied eight transfer functions in our BGWO and compared their performance. The test functions whose minimum points are gotten at 0 are selected as bench functions but also those functions their minimum value are gotten at 1.

Game Analysis of Inter-Organizational Knowledge Spillovers in Agricultural Supply Chain

In this research, Shuffled frog-leaping algorithm (SFLA) has long been considered as new evolutionary algorithm of group evolution, and has a high computing performance and excellent ability for global search. Knapsack problem is a typical NP-complete problem. For the discrete search space, this paper presents the improved SFLA, and solves the knapsack problem by using the algorithm. Experimental results show the feasibility and effectiveness of this method.

Continuous fitness landscape analysis using a chaos-based random walk algorithm

Extensive research on heuristic algorithms has proved their potential in solving complex optimization problems. However, it is not easy to choose the best heuristic technique for solving a particular problem. Fitness landscape analysis is used for understanding the problem characteristics based on which the best-suited algorithm for the problem can be chosen. Compared to the literature on discrete search spaces, only a few significant works have been undertaken on landscape analysis in continuous search spaces. Random walk (RW) algorithm has been used for generating sample points in the search space, and fitness landscape is created based on the relative fitness of the neighboring sample points. This paper proposes a chaos-based random walk algorithm, called as the chaotic random walk (CRW), applied in continuous search space to generate the landscape structure for a problem. The chaotic map is used to generate the chaotic pseudorandom numbers for determining variable scaled step size and direction of the proposed RW algorithm. Histogram analysis demonstrates better coverage of search space by the CRW algorithm compared to the simple and progressive random walk algorithms. In addition, we test the efficiency of the proposed method by quantifying the ruggedness and deception of a problem using entropy and fitness distance correlation measures. Experiments are conducted on the IEEE Congers on Evolutionary Computing 2013 benchmark functions in continuous search space having different levels of complexity. Extensive experiments indicate the capability for generating landscape structure on the continuous search space and efficiency of the proposed method to investigate the structural features of fitness landscapes.

An evolutionary strategy based on partial imitation for solving optimization problems

In this work we introduce an evolutionary strategy to solve combinatorial optimization tasks, i.e. problems characterized by a discrete search space. In particular, we focus on the Traveling Salesman Problem (TSP), i.e. a famous problem whose search space grows exponentially, increasing the number of cities, up to becoming NP-hard. The solutions of the TSP can be codified by arrays of cities, and can be evaluated by fitness, computed according to a cost function (e.g. the length of a path). Our method is based on the evolution of an agent population by means of an imitative mechanism, we define `partial imitation'. In particular, agents receive a random solution and then, interacting among themselves, may imitate the solutions of agents with a higher fitness. Since the imitation mechanism is only partial, agents copy only one entry (randomly chosen) of another array (i.e. solution). In doing so, the population converges towards a shared solution, behaving like a spin system undergoing a cooling process, i.e. driven towards an ordered phase. We highlight that the adopted `partial imitation' mechanism allows the population to generate solutions over time, before reaching the final equilibrium. Results of numerical simulations show that our method is able to find, in a finite time, both optimal and suboptimal solutions, depending on the size of the considered search space.

Feature selection using Forest Optimization Algorithm

Feature selection as a combinatorial optimization problem is an important preprocessing step in data mining; which improves the performance of the learning algorithms with the help of removing the irrelevant and redundant features. As evolutionary algorithms are reported to be suitable for optimization tasks, so Forest Optimization Algorithm (FOA) – which is initially proposed for continuous search problems – is adapted to be used for feature selection as a discrete search space problem. As the result, Feature Selection using Forest Optimization Algorithm (FSFOA) is proposed in this article in order to select the more informative features from the datasets. The proposed FSFOA is validated on several real world datasets and it is compared with some other methods including HGAFS, PSO and SVM-FuzCoc. The results of the experiments show that, FSFOA can improve the classification accuracy of classifiers in some selected datasets. Also, we have compared the dimensionality reduction of the proposed FSFOA with other available methods.

Designing Noise-Minimal Rotorcraft Approach Trajectories

NASA and the international aviation community are investing in the development of a commercial transportation infrastructure that includes the increased use of rotorcraft, specifically helicopters and civil tilt rotors. However, there is significant concern over the impact of noise on the communities surrounding the transportation facilities. One way to address the rotorcraft noise problem is by exploiting powerful search techniques coming from artificial intelligence to design low-noise flight profiles that can be then validated though field tests. This article investigates the use of discrete heuristic search methods to design low-noise approach trajectories for rotorcraft. Our work builds on a long research tradition in trajectory optimization using either numerical methods or discrete search. Novel features of our approach include the use of a discrete search space with a resolution that can be varied, and the coupling of search with a robust simulator to evaluate candidates. The article includes a systematic comparison of different search techniques; in particular, in the experiments, we are able to do a trade study that compares complete search algorithms such as A * with faster but approximate methods such as local search.

A dynamic-edge ACS algorithm for continuous variables problems

Ant colony systems (ACS) have been successfully applied to solving optimization problems. Especially, they are efficient and effective in finding nearly optimal solutions to discrete search spaces. When the solution spaces of the problems to be solved are continuous, it is not so appropriate to use the original ACS to solve it. This paper thus proposes a dynamic-edge ACS algorithm for solving continuous variables problems. It can dynamically generate edges between two nodes and give a pheromone measures for them in a continuous solution space through distribution functions. In addition, it maps the encoding representation and the operators of the original ACS into continuous spaces easily. The proposed algorithm thus provides a simple and standard approach for applying ACS to a problem that has a continuous solution space, and retains the original process and characteristics of the traditional ACS. Several constrained functions are also evaluated to demonstrate the performance of the proposed algorithm.

Maximizing the spread of positive influence in signed social networks

Influence maximization in a social network involves identifying an initial subset of nodes with a pre-defined size in order to begin the information diffusion with the objective of maximizing the influenced nodes. In this study, a sign-aware cascade (SC) model is proposed for modeling the effect of both trust and distrust relationships on activation of nodes with positive or negative opinions towards a product in the signed social networks. It is proved that positive influence maximization is NP-hard in the SC model and influence function is neither monotone nor submodular. For solving this NP-hard problem, a particle swarm optimization (PSO) method is presented which applies the random keys representation technique to convert the continuous search space of the PSO to the discrete search space of this problem. To improve the performance of this PSO method against premature convergence, a re-initialization mechanism for portion of particles with poorer fitness values and a heuristic mutation operator for global best particle are proposed. Experiments establish the effectiveness of the SC in modeling the real-world cascades. In addition, PSO method is compared with the well-known algorithms in the literature on two real- world data sets. The evaluation results demonstrate that the proposed method outperforms the compared algorithms significantly in the SC model.

Learning programs from noisy data

We present a new approach for learning programs from noisy datasets. Our approach is based on two new concepts: a regularized program generator which produces a candidate program based on a small sample of the entire dataset while avoiding overfitting, and a dataset sampler which carefully samples the dataset by leveraging the candidate program's score on that dataset. The two components are connected in a continuous feedback-directed loop. We show how to apply this approach to two settings: one where the dataset has a bound on the noise, and another without a noise bound. The second setting leads to a new way of performing approximate empirical risk minimization on hypotheses classes formed by a discrete search space. We then present two new kinds of program synthesizers which target the two noise settings. First, we introduce a novel regularized bitstream synthesizer that successfully generates programs even in the presence of incorrect examples. We show that the synthesizer can detect errors in the examples while combating overfitting -- a major problem in existing synthesis techniques. We also show how the approach can be used in a setting where the dataset grows dynamically via new examples (e.g., provided by a human). Second, we present a novel technique for constructing statistical code completion systems. These are systems trained on massive datasets of open source programs, also known as ``Big Code''. The key idea is to introduce a domain specific language (DSL) over trees and to learn functions in that DSL directly from the dataset. These learned functions then condition the predictions made by the system. This is a flexible and powerful technique which generalizes several existing works as we no longer need to decide a priori on what the prediction should be conditioned (another benefit is that the learned functions are a natural mechanism for explaining the prediction). As a result, our code completion system surpasses the prediction capabilities of existing, hard-wired systems.

Discrete particle swarm optimization method for the large-scale discrete time–cost trade-off problem

Despite many research studies have concentrated on designing heuristic and meta-heuristic methods for the discrete time–cost trade-off problem (DTCTP), very little success has been achieved in solving large-scale instances. This paper presents a discrete particle swarm optimization (DPSO) to achieve an effective method for the large-scale DTCTP. The proposed DPSO is based on the novel principles for representation, initialization and position-updating of the particles, and brings several benefits for solving the DTCTP, such as an adequate representation of the discrete search space, and enhanced optimization capabilities due to improved quality of the initial swarm. The computational experiment results reveal that the new method outperforms the state-of-the-art methods, both in terms of the solution quality and computation time, especially for medium and large-scale problems. High quality solutions with minor deviations from the global optima are achieved within seconds, for the first time for instances including up to 630 activities. The main contribution of the proposed particle swarm optimization method is that it provides high quality solutions for the time–cost optimization of large size projects within seconds, and enables optimal planning of real-life-size projects.